Selection of polishing parameters to generate removal profile

ABSTRACT

Values are selected for a plurality of controllable parameters of a chemical mechanical polishing system that includes a carrier head with a plurality of zones to apply independently controllable pressures on a substrate. Data is stored relating variation in removal profile on a front surface of the substrate to variation in the controllable parameters, the data including removal at a plurality of positions on the front surface of the substrate, there being a greater number of positions than chambers. A value is determined for each parameter of the plurality of controllable parameters to minimize a difference between a target removal profile and an expected removal profile calculated from the data relating variation in removal profile on a front surface of the substrate to variation in the parameters. The value for each parameter of the plurality of controllable parameters is stored.

TECHNICAL FIELD

The present disclosure relates to control of a carrier head duringchemical mechanical polishing.

BACKGROUND

An integrated circuit is typically formed on a substrate by thesequential deposition of conductive, semiconductive, or insulativelayers on a silicon wafer. One fabrication step involves depositing afiller layer over a non-planar surface and planarizing the filler layer.For certain applications, the filler layer is planarized until the topsurface of a patterned layer is exposed. A conductive filler layer, forexample, can be deposited on a patterned insulative layer to fill thetrenches or holes in the insulative layer. After planarization, theportions of the conductive layer remaining between the raised pattern ofthe insulative layer form vias, plugs, and lines that provide conductivepaths between thin film circuits on the substrate. For otherapplications, such as oxide polishing, the filler layer is planarizeduntil a predetermined thickness is left over the non planar surface. Inaddition, planarization of the substrate surface is usually required forphotolithography.

Chemical mechanical polishing (CMP) is one accepted method ofplanarization. This planarization method typically requires that thesubstrate be mounted on a carrier or polishing head. The exposed surfaceof the substrate is typically placed against a rotating polishing pad.The carrier head provides a controllable load on the substrate to pushit against the polishing pad. An abrasive polishing slurry is typicallysupplied to the surface of the polishing pad.

A problem in CMP is achieving a uniform thickness of the substratelayer. Variations in the slurry distribution, the polishing padcondition, the relative speed between the polishing pad and thesubstrate, and the dynamics of the interaction of the polishing pad andthe substrate, can cause variations in the material removal rate acrossthe substrate. These variations, as well as variations in the initialthickness of the substrate layer, cause variations in resultingthickness of the substrate layer.

Some carrier heads include multiple independently pressurizablechambers. The chambers can provide independently controllable pressureson different portions of the substrate. By providing different pressuresto the different chambers, the pressure on corresponding portions of thesubstrate, and thus the polishing rate on those corresponding portions,can be selected to partially compensate for non-uniformity in thesubstrate layer.

SUMMARY

Some control systems adjust the pressure in the chambers of a carrierhead to achieve a target polishing rate profile under an assumption thata uniform pressure is applied across the region of the substrate coveredby the chamber. For example, the target polishing rate profile mayinclude only a single target value per chamber, or an algorithm thatthat calculates a set of pressures for the chambers can compare the samevalue for a given chamber to the target value for each point on thewafer below the given chamber.

However, in actual operation, the pressure applied by a chamber can benon-uniform across the region of the substrate covered by the chamber.In addition, the pressure in one chamber can influence the pressure onregions of the substrate that are not directly under the chamber, e.g.,the pressure in one chamber can “spill over” to regions covered by otherchambers. A controller that calculates the pressures for the chamberscan account for this spill-over and non-uniformity. For example, aremoval profile that is actually generated by a chamber can be measured,and this removal profile can be used when calculating the set ofpressures for the chambers. More generally, removal profiles can begenerated for process parameters at different values, and these removalprofiles can be used when calculating values for the process parametersto achieve a target profile.

In one aspect, there is a method of selecting values for a plurality ofcontrollable parameters of a chemical mechanical polishing system thatincludes a carrier head with a plurality of zones to apply independentlycontrollable pressures on a substrate. Data is stored relating variationin removal profile on a front surface of the substrate to variation inthe controllable parameters, the data including removal at a pluralityof positions on the front surface of the substrate, there being agreater number of positions than chambers. A value is determined foreach parameter of the plurality of controllable parameters to minimize adifference between a target removal profile and an expected removalprofile calculated from the data relating variation in removal profileon a front surface of the substrate to variation in the parameters. Thevalue for each parameter of the plurality of controllable parameters isstored.

Implementations may optionally include one or more of the followingfeatures. The plurality of controllable parameters may include pressuresfor a plurality of chambers in the carrier head that apply pressure tothe plurality of zones. The plurality of controllable parameters mayinclude a pressure for a chamber in the carrier head that appliespressure to a retaining ring of the carrier head. The plurality of zonesmay be arranged concentrically and the plurality of positions may beradial distances from the center of the substrate. The plurality ofpositions may include a first plurality of positions below a first zoneof the plurality of zones and a second plurality of positions below asecond zone of the plurality of zones. The plurality of controllableparameters may include a platen rotation rate or a carrier head rotationrate. The positions may be regularly spaced across the substrate. Theremay be a greater number of positions than parameters. Storing data mayinclude storing a plurality of measured removal profiles, and eachmeasured removal profile may include an amount removed from a frontsurface of a substrate at each of a plurality of positions on thesubstrate. The plurality of removal profiles may include a baselineremoval profile for one substrate polished under a set of baselinevalues for the plurality of controllable parameters. The plurality ofremoval profiles may include a plurality of adjusted removal profilesfor additional substrates, and each additional substrate may be polishedwith exactly one of the plurality of controllable parameters set to anadjusted value that is different from the baseline value for the onesubstrate. The one of the plurality of controllable parameters may bedifferent for each additional substrate. Calculating the expectedremoval profile may include calculatingERP=BRP+[K][P′]*BRPwhere ERP is the expected removal profile, BRP is the baseline removalprofile, [K] is a matrix of constants having a number of columns equalto the total number of parameters, N, and a number of rows equal to thetotal number of radial positions on the substrate surface where theprofile is measured, M, and [P′] is a column matrix of adjustedpolishing system parameters. The adjusted polishing system parametersmay be

$P_{i}^{\prime} = \frac{P_{i} - {P\; 0_{i}}}{P\; 0_{i}}$where P_(i) is the value of the polishing parameter to be calculated,and P0_(i) is the baseline value for the polishing parameter. The matrix[K] may be expressed as

$\lbrack K\rbrack = \begin{bmatrix}K_{1,1} & \ldots & K_{1,N} \\\vdots & \ddots & \vdots \\K_{M,1} & \ldots & K_{M,N}\end{bmatrix}$where M is the total number of radial substrate positions x at whichprofile measurements are taken and N is the total number of parameters.Constants of the matrix may be

$K_{x,i} = \frac{{ARP}_{x,i} - {BRP}_{x}}{{BRP}_{x}*\left\lbrack \frac{{PA}_{i} - {P\; 0_{i}}}{P\; 0_{i}} \right\rbrack}$where K_(x,i) is the value of the matrix [K] corresponding to thesubstrate profile position x for the i^(th) parameter, ARP_(x,i) is theamount of material removed at radial position x for the i^(th) adjustedparameter, BRP_(x) is the amount of material removed at radial positionx according to the previously calculated baseline removal profile,PA_(i) is the i^(th) parameter value used to generate the i^(th)adjusted removal profile, and P0_(i) is the i^(th) parameter value forthe baseline removal profile. Determining a value for each parameter mayinclude iteratively adjusting at least one value, calculating theexpected removal profile from the data, and determining a differencebetween the expect pressure profile and the target pressure profile.Determining the value for the chamber pressure for each chamber mayinclude Newton's method of optimization. The target removal profile maybe generated from data collected in-situ during polishing of a firstsubstrate, and adjusting at least one pressure applied during thepolishing of the first substrate to match the value for the chamberpressure. The target removal profile may be generated from datacollected in-situ during polishing of a first substrate at a firstplaten, and polishing the first substrate at a different second platenusing the chamber pressure. The target removal profile may be generatedfrom data collected in-situ during polishing of a first substrate at afirst platen, and polishing a different second substrate at the firstplaten using the chamber pressure.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other aspects,features, and advantages will be apparent from the description anddrawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic cross-sectional view of an example chemicalmechanical polishing apparatus.

FIG. 2 shows a schematic top view of a substrate having multiple zones.

FIG. 3 shows a top view of a polishing pad and shows locations wherein-situ measurements can be taken on a substrate.

FIG. 4 shows an aggregation of measured pressure distributions resultingfrom pressurizing respective carrier head chambers one at a time.

FIG. 5 shows a collection of measured pressure distributions resultingfrom pressurizing a carrier head chamber to various pressures whileadjacent chambers are unpressurized.

FIG. 6 shows a collection of measured pressure distributions resultingfrom pressurizing a carrier head chamber to various pressures whileadjacent chambers are pressurized to a baseline pressure.

FIG. 7 is a flow chart of a method of generating a pressure distributionprofile.

FIG. 8 is a flow chart of a method of selecting pressures for chambersin a carrier head.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

A polishing system can have multiple controllable parameters thatinfluence the polishing rate on of substrate. Each parameter controlsthe operation of an associated hardware component of the polishingsystem, and the parameter can be set by software, e.g., in a controlsystem. Examples of controllable parameters include pressure in chambersin a carrier head (including pressure in chambers that apply downwardpressure to the substrate or to a retaining ring), carrier head rotationrate, and platen rotation rate.

FIG. 1 illustrates an example of a polishing apparatus 100. Thepolishing apparatus 100 includes a rotatable disk-shaped platen 120 onwhich a polishing pad 110 is situated. The platen is operable to rotateabout an axis 125. For example, a motor 121 can turn a drive shaft 124to rotate the platen 120. The polishing pad 110 can be detachablysecured to the platen 120, for example, by a layer of adhesive. Thepolishing pad 110 can be a two-layer polishing pad with an outerpolishing layer 112 and a softer backing layer 114.

The polishing apparatus 100 can include a combined slurry/rinse arm 130.During polishing, the arm 130 is operable to dispense a polishing liquid132, such as a slurry, onto the polishing pad 110. While only oneslurry/rinse arm 130 is shown, additional nozzles, such as one or morededicated slurry arms per carrier head, can be used. The polishingapparatus can also include a polishing pad conditioner to abrade thepolishing pad 110 to maintain the polishing pad 110 in a consistentabrasive state.

The polishing apparatus 100 can further include a carrier heads 140. Thecarrier head 140 may be operable to hold a substrate 10 against thepolishing pad 110. While only one carrier head 140 is shown, additionalcarrier heads can be used and may be preferable is some implementations.In such embodiments, each carrier head 140 can be controlledindependently, with the polishing parameters associated with the carrierhead (e.g., chamber pressure, retaining ring pressure, and/or carrierhead rotation rate), independently set.

Carrier head 140 can include a retaining ring 142 to retain thesubstrate 10 below a flexible membrane 144. Carrier head 140 may alsoinclude a plurality of independently controllable pressurizable chambersdefined by the membrane, e.g., 3 chambers 146 a-146 c, which can applyindependently controlled amounts of pressure to associated zones 148a-148 c on the flexible membrane 144, and thus on the back side of thesubstrate 10 (see FIG. 2). The flexible membrane 144 can be formed of anelastic material, such as a high strength silicone rubber. Referring toFIG. 2, the center zone 148 a can be substantially circular, and theremaining zones 148 b-148 e can be concentric annular zones around thecenter zone 148 a. Although only three chambers are illustrated in FIGS.1 and 2 for ease of illustration, there could be two chambers, or fouror more chambers, e.g., five chambers.

Returning to FIG. 1, carrier head 140 is suspended from a supportstructure 150, e.g., a carousel, and is connected by a drive shaft 152to a carrier head rotation motor 154 so that the carrier head can rotateabout an axis 155. Optionally, carrier head 140 can oscillate laterally,e.g., on sliders on the carousel 150, or by rotational oscillation ofthe carousel itself. In operation, the platen is rotated about itscentral axis 125, and the carrier head is rotated about its central axis155 and translated laterally across the top surface of the polishingpad.

Again, while only one carrier head 140 is shown, more carrier heads canbe provided to hold additional substrates so that the surface area ofpolishing pad 110 may be used efficiently. Thus, the number of carrierhead assemblies adapted to hold substrates for a simultaneous polishingprocess can be based, at least in part, on the surface area of thepolishing pad 110.

The polishing apparatus also includes an in-situ monitoring system 160,the data from which can be used by a controller 190 to determine whetherto adjust a polishing rate or an adjustment for the polishing rate asdiscussed below. The in-situ monitoring system 160 can include anoptical monitoring system, e.g., a spectrographic monitoring system, oran eddy current monitoring system.

In one embodiment, the monitoring system 160 is an optical monitoringsystem. An optical access through the polishing pad is provided byincluding an aperture (i.e., a hole that runs through the pad) or asolid window 118. The solid window 118 can be secured to the polishingpad 110, e.g., as a plug that fills an aperture in the polishing pad,e.g., is molded to or adhesively secured to the polishing pad, althoughin some implementations the solid window can be supported on the platen120 and project into an aperture in the polishing pad.

The optical monitoring system 160 can include a light source 162, alight detector 164, and circuitry 166 for sending and receiving signalsbetween a remote controller 190, e.g., a computer, and the light source162 and light detector 164. One or more optical fibers can be used totransmit the light from the light source 162 to the optical access inthe polishing pad, and to transmit light reflected from the substrate 10to the detector 164. For example, a bifurcated optical fiber 170 can beused to transmit the light from the light source 162 to the substrate 10and back to the detector 164. The bifurcated optical fiber an include atrunk 172 positioned in proximity to the optical access, and twobranches 174 and 176 connected to the light source 162 and detector 164,respectively.

In some implementations, the top surface of the platen can include arecess 128 into which is fit an optical head 168 that holds one end ofthe trunk 172 of the bifurcated fiber. The optical head 168 can includea mechanism to adjust the vertical distance between the top of the trunk172 and the solid window 118.

The output of the circuitry 166 can be a digital electronic signal thatpasses through a rotary coupler 129, e.g., a slip ring, in the driveshaft 124 to the controller 190 for the optical monitoring system.Similarly, the light source can be turned on or off in response tocontrol commands in digital electronic signals that pass from thecontroller 190 through the rotary coupler 129 to the optical monitoringsystem 160. Alternatively, the circuitry 166 could communicate with thecontroller 190 by a wireless signal.

The light source 162 can be operable to emit white light. In oneimplementation, the white light emitted includes light havingwavelengths of 200-800 nanometers. A suitable light source is a xenonlamp or a xenon mercury lamp.

The light detector 164 can be a spectrometer. A spectrometer is anoptical instrument for measuring intensity of light over a portion ofthe electromagnetic spectrum. A suitable spectrometer is a gratingspectrometer. Typical output for a spectrometer is the intensity of thelight as a function of wavelength (or frequency).

As noted above, the light source 162 and light detector 164 can beconnected to a computing device, e.g., the controller 190, operable tocontrol their operation and receive their signals. The computing devicecan include a microprocessor situated near the polishing apparatus,e.g., a programmable computer. With respect to control, the computingdevice can, for example, synchronize activation of the light source withthe rotation of the platen 120. The controller 190 can also generate atarget pressure profile based on the data from the optical monitoringsystem 160, store the target pressure profile, and calculate a set ofpressures for the chambers in the carrier head to achieve the targetpressure profile.

In some implementations, the light source 162 and detector 164 of thein-situ monitoring system 160 are installed in and rotate with theplaten 120. In this case, the motion of the platen will cause the sensorto scan across each substrate. In particular, as the platen 120 rotates,the controller 190 can cause the light source 162 to emit a series offlashes starting just before and ending just after each substrate 10passes over the optical access. Alternatively, the computing device cancause the light source 162 to emit light continuously starting justbefore and ending just after each substrate 10 passes over the opticalaccess. In either case, the signal from the detector can be integratedover a sampling period to generate spectra measurements at a samplingfrequency.

In operation, the controller 190 can receive, for example, a signal thatcarries information describing a spectrum of the light received by thelight detector for a particular flash of the light source or time frameof the detector. Thus, this spectrum is a spectrum measured in-situduring polishing.

As shown by in FIG. 3, if the detector is installed in the platen, dueto the rotation of the platen (shown by arrow 204), as the window 108travels below one carrier head (e.g., the carrier head holding the firstsubstrate 10 a), the optical monitoring system making spectrameasurements at a sampling frequency will cause the spectra measurementsto be taken at locations 201 in an arc that traverses the firstsubstrate 10 a. For example, each of points 201 a-201 k represents alocation of a spectrum measurement by the monitoring system of the firstsubstrate 10 a (the number of points is illustrative; more or fewermeasurements can be taken than illustrated, depending on the samplingfrequency). As shown, over one rotation of the platen, spectra areobtained from different radii on the substrate 10 a. That is, somespectra are obtained from locations closer to the center of thesubstrate 10 a and some are closer to the edge.

As mentioned above with reference to FIG. 1, pressurization of thechambers 146 a-146 c controls the downward pressure of the substrate 10against the polishing pad 110. In general, the majority of the pressureapplied by a chamber to an associated zone of the membrane is impartedon a corresponding zone of the front surface of the substrate againstthe polishing pad. Some amount of the applied pressure, however, can“spill over” between zones on the front surface of the substrate (seeFIG. 4). In addition, the pressure applied by the chamber within theassociated zone may not be completely uniform. FIG. 4 illustratesseveral pressure distribution curves, each of which is an illustrativegraph of the pressure applied at the front surface of the substrate as afunction of the distance from the center of the substrate, assuming thata particular chamber in the carrier head is pressurized. Referring toFIG. 4, in this example, as shown by pressure distribution curve 200 a,at least some pressure provided by chamber 146 a is applied to bothzones 148 a and 148 b of substrate 10. Likewise, as shown by pressuredistribution curve 200 b, at least some pressure provided by chamber 146b is applied to each of zones 148 a-148 c, and as shown by pressuredistribution curve 200 c, at least some pressure provided by chamber 146c is applied to zones 148 c and 148 b. In addition, even within zone 148a of substrate 10, the pressure applied by chamber 146 a is notcompletely uniform.

Without being limited to any particular theory, the pressure spill overand non-uniformity may be a result of the flexible nature of membrane144. That is, under pressure, portions of membrane 144 may be stretchedor forced to expand beyond the originally defined zones. Further, andagain without being limited to any particular theory, the pressure spillover may be caused by the thickness of the substrate 10. For example,the substrate 10 may be sufficiently thick such that pressure applied tothe back surface (i.e., the substrate surface contacting the membrane144) spreads out radially across the zone boundaries as it propagates tothe front surface (i.e., the substrate surface contacting the polishingpad).

In general, a polishing system is controlled so that the post-polishedsubstrate has a target surface profile, e.g., a target thickness acrossthe substrate surface. The target surface profile may be uniform (e.g.,planar) or non-uniform across the substrate surface. The target surfaceprofile can be set manually by a manufacturer of the polishing system,set manually by an operator of the polishing system, e.g., an employeeof the semiconductor fab, or could be generated automatically bycomputer software based on measurements of the performance of othertools at the semiconductor fab, e.g., to compensate for non-uniformdeposition or removal by the other tools.

As noted above, the polishing rate (i.e., the rate at which material isremoved from the substrate 10) can vary according to several polishingsystem parameters, e.g., chamber pressure retaining ring pressure,carrier head rotation rate, platen rotation rate, etc. As such, thecombination of such parameters can determine the surface profile of thesubstrate 10 after polishing.

The controller 190 can be configured to store a target removal profile.The controller 190 uses the target removal profile to set the processparameters for the polishing system 20, e.g., the pressures in thechambers 146 a-146 c of the carrier head 140. The target removal profilerepresents a desired amount of material to be removed across the frontsurface of the substrate. In some implementations, the target removalprofile can be determined, e.g., by the controller 190 or anothercomputer system which forwards the target removal profile to thecontroller 190, based on an initial surface profile of the substrate 10(e.g., as measured at a metrology station prior to polishing or by anin-situ monitoring system). For example, the target removal profile canbe calculated asTRP=ISP−TSP  (1)where TRP is the target removal profile, ISP is the initial surfaceprofile, and TSP is the target surface profile. Alternatively, if theinitial surface profile is unknown, then the initial surface profile canbe set to a default value. Also, in some situations, the operator of thepolishing system may desire to remove a target amount from the substraterather than achieve a target profile. In addition, in some situations,the operator of the polishing system may simply set the target removalprofile, e.g., based on a priori principals or previous experience. Inthis last case, the user can generate the target removal profile by userinput, e.g., to the controller 190 or to another computer system whichforwards the target removal profile to the controller 190.

The controller 190 can generate and store an expected removal profile(that is, the amount of material expected to be removed across the frontsurface of substrate 10) for a given set of values for the processparameters.

In general, it is advantageous to select values for the processparameters, e.g., the pressures in the chambers of the carrier head, sothat the expected removal profile closely approximates (includingexactly matches) the target removal profile. For example, if theexpected removal profile closely approximates the target removalprofile, then the actual removal profile, i.e., the amount actuallyremoved from a substrate in polishing, should also closely approximatethe target removal profile.

It should be noted that the removal profile described herein may beexpressed as an amount of material removed from the substrate surfaceor, equivocally, as a material removal rate (e.g., by dividing removalprofile by an actual or expected time spent polishing). Further, itshould also be understood that the data used for the calculation couldbe stored in various different units, so long the data is converted intoconsistent units for the purpose of calculation.

In some implementations, the expected removal profile can be calculatedas a linear function of the polishing parameters, e.g., the pressuresprovided by the chambers 146 a-146 c. For example, increasing thepressure in a particular chamber can result in a linear increase in thepressure of the substrate on the polishing pad, and thus a linearincrease in the polishing rate, and thus a linear increase in the amountremoved. In particular, for a plurality of different points on thesubstrate, e.g., points at different radial distances from the center ofthe substrate, the removal can be calculated as a linear combination ofthe polishing parameters, e.g., the pressures in the chambers.

A relationship between the polishing parameters and the removal profilecan be determined based on measured data. For example, removal profilescan be measured for test substrates, which each test substrate beingpolished using a different set of values for the polishing parameters.In particular, one substrate can be polished under a set of baselinevalues for the polishing parameters, and a baseline removal profile ismeasured from the one substrate. Then, for each parameter that is to beset by the process, an additional substrate is polished, with thatparameter set to an adjusted value that is different from the baselinevalue (but the other parameters set to their baseline values), and anadjusted removal profile is measured from the additional substrate.

In some implementations, the expected removal profile can be calculatedasERP=BRP+[K][P′]*BRP  (2)where ERP is the expected removal profile, BRP is a baseline removalprofile, [K] is a matrix of constants having a number of columns equalto the total number of parameters, N, and a number of rows equal to thetotal number M of radial positions on the substrate surface where theprofile is measured, and [P′] is a vector of adjusted polishing systemparameters. The total number of positions M can be greater than thenumber of chambers in the carrier head, and can be greater than thetotal number of controllable parameters N.

The baseline removal profile can be determined by polishing a test waferat a set of default polishing parameters, P0₁, P0₂, . . . P0_(N), andcomparing a measured (e.g., at a metrology station) surface profileafter polishing to the initial surface profile. For example, thebaseline removal profile can be calculated asBRP=ISP−BSP  (3)where BRP is the baseline removal profile, ISP is the initial surfaceprofile, and BSP is the baseline surface profile achieved as a result ofpolishing a test substrate at baseline parameters P0₁, P0₂, . . .P0_(N).

The constants of the matrix [K] can be determined based on the baselineremoval profile and the adjusted removal profiles. As noted above, theadjusted removal profiles can be generated by polishing an additionalsubstrate for each parameter while modifying that parameter, onesubstrate at a time. For example, an adjusted removal profile can becalculated for each modified parameter according to the equationARP _(i) =ISP−ASP _(i)  (4)where ARP_(i) is the adjusted removal profile resulting from adjustmentof the i^(th) polishing parameter, ISP is the initial surface profile,and ASP_(i) is the adjusted surface profile measured as a result ofadjusting the i^(th) parameter. In particular, the constant values ofthe matrix [K] can be calculated as

$\begin{matrix}{K_{x,i} = \frac{{ARP}_{x,i} - {BRP}_{x}}{{BRP}_{x}*\left\lbrack \frac{{PA}_{i} - {P\; 0_{i}}}{P\; 0_{i}} \right\rbrack}} & (5)\end{matrix}$where K_(x,i) is the value of the matrix [K] corresponding to thesubstrate profile position x for the i^(th) parameter, ARP_(x,i) is theamount of material removed at radial position x for the i^(th) adjustedparameter, BRP_(x) is the amount of material removed at radial positionx according to the previously calculated baseline removal profile,PA_(i) is the i^(th) parameter value used to generate the i^(th)adjusted removal profile, and P0_(i) is the i^(th) parameter value forthe baseline removal profile. As such, the matrix [K] can be expressedas

$\begin{matrix}{\lbrack K\rbrack = \begin{bmatrix}K_{1,1} & \ldots & K_{1,N} \\\vdots & \ddots & \vdots \\K_{M,1} & \ldots & K_{M,N}\end{bmatrix}} & (6)\end{matrix}$where M is the total number of radial substrate positions x at whichprofile measurements are taken and N is the total number of parameters.The variable expressions of the vector [P′], P₁′, P₂′ . . . P_(N)′ aredefined by

$\begin{matrix}{P_{i}^{\prime} = \frac{P_{i} - {P\; 0_{i}}}{P\; 0_{i}}} & (7)\end{matrix}$where P_(i)′ is the value of the vector [P′] corresponding to the i^(th)parameter, P0_(i) is the i^(th) parameter value for the baseline removalprofile, and P_(i) is the variable proposed parameter value. Thus, theexpected removal profile can be expressed as

$\begin{matrix}{{ERP} = {{BRP} + {{\begin{bmatrix}K_{1,1} & \ldots & K_{1,N} \\\vdots & \ddots & \vdots \\K_{M,1} & \ldots & K_{M,N}\end{bmatrix}\begin{bmatrix}P_{1}^{\prime} \\\vdots \\P_{N}^{\prime}\end{bmatrix}}*{BRP}}}} & (8)\end{matrix}$where ERP is the expected removal profile, BRP is the baseline removalprofile, and values for the matrix [K] and the vector [P′] arecalculated as described above with reference to equations (5) and (7).

In some implementations, the various profiles generated fromexperimental measurement, e.g., the base removal profile and theadjusted removal profile, includes measured removal values for aplurality of positions under each zone, e.g., a plurality of positionsunder each chamber. Similarly, the calculated expected removal includescalculated values for the expected removal at plurality of positionsunder each zone, e.g., a plurality of positions under each chamber. Insome implementations, the profiles, e.g., the base removal profile, theadjusted removal profile and expected removal profile, includes valuesfor twenty to three hundred positions on the substrate. For example, theprofiles can include values for positions on the substrate at a regularspacing of 1 mm.

In some implementations, a suitable combination of parameters can bedetermined by solving for respective parameters that minimize adifference between the expected removal profile and the target removalprofile. That is, the value of ERP−TRP is minimized. In many cases, oneor more distinct sets of parameters can be determined mathematicallythat provide an expected removal profile equaling the target removalprofile, such that ERP−TRP=0. In other cases a difference value betweenthe expected removal profile and the target removal profile can beminimized to an acceptable, non-zero value by using the followingequationΔ=Σ_(x=1) ^(M)(ERP _(x) −TRP _(x))²  (9)where Δ is a difference value, ERP_(x) is the variable value of theexpected removal profile at a radial position x on the substrate andTRP_(x) is the constant value of the target removal profile at theradial position x. Values of P_(i) can be calculated to find an expectedremoval profile that closely approximates the target removal profile.

In some examples, an iterative process (e.g., Newton's method ofoptimization) may be used in conjunction with equation (9) to determinean appropriate combination of parameter values to find a minimumdifference between the estimated pressure profile and the targetpressure profile, e.g., to find a minimum value for Δ. For example, acommercially available solver function (e.g., the Microsoft Excel solverfunction, the MATLAB solver function, and/or the Wolfram Mathematicasolver function, etc.) may be used to determine an appropriatecombination of parameter values. In some cases, there may be severalsuitable combinations of parameter values.

Once an appropriate set of parameter values has been determined, asdescribed above, the controller 190 can be configured to store theparameter values and execute a polishing technique accordingly toprovide one or more substrates having the target surface profile.

In most cases, when all other parameter values are to be held constant,the polishing rate at a particular location on the substrate 10 can varylinearly with the pressure applied to the substrate, at that location ofthe substrate, on the polishing pad.

In another implementation, the controller 190 can be configured to storea target pressure profile. The controller 190 uses the target pressureprofile to set the pressures in the chambers 146 a-146 c of the carrierhead 140. The target pressure profile represents a desired pressure tobe applied by the front surface of the substrate on the polishing pad.In some implementations, the target pressure profile can be consideredthe pressure distribution that should produce the polishing ratesnecessary to achieve the target surface profile. In someimplementations, the target pressure profile can be determined, e.g., bythe controller 190 or another computer system which forwards the targetpressure profile to the controller 190, based on the initial surfaceprofile of the substrate 10 (e.g., as measured at a metrology stationprior to polishing or by an in-situ monitoring system) and the knownlinear relationship between polishing rate and polishing pressure. Thatis, the target pressure profile can be calculated asTPP=m*(ISP−TSP)  (10)where TPP is the target pressure profile, ISP is the initial surfaceprofile, TSP is the target surface profile, and m is an empiricallydetermined constant. Alternatively, if the initial surface profile isunknown, then the initial surface profile can be set to a default value.Also, in some situations, the operator of the polishing system maydesire to remove a target amount from the substrate rather than achievea target profile. In this case, the target pressure profile is simply alinear function of the target amount to remove. In addition, in somesituations, the operator of the polishing system may simply set thetarget polishing profile, e.g., based on a priori principals or previousexperience. In this last case, the user can generate the target pressureprofile by user input, e.g., to the controller 190 or to anothercomputer system which forwards the target pressure profile to thecontroller 190.

The controller 190 can generate and store an expected pressure profile(that is, the pressure distribution across the front surface ofsubstrate 10 on the polishing pad that is expected when chambers 146a-146 c are pressurized). In some implementations, the target pressureprofile can be approximated by a pressure profile expected to be appliedby the chambers 146 a-146 c. Generally, it is advantageous to closelyapproximate the target pressure profile with the expected pressureprofile. For example, closely approximating the target pressure profilewith the expected pressure profile may result in an actual front surfaceprofile of the substrate 10 that is a close approximation of the targetsurface profile after polishing.

In some implementations, the expected pressure applied by the substrateto the polishing pad can be calculated as a linear function of thepressure provided by the chambers 146 a-146 c. For example, increasingthe pressure in a particular chamber can result in a linear increase inthe pressure of the substrate on the polishing pad, and thus a linearincrease in the polishing rate. In particular, for a plurality ofdifferent points on the substrate, e.g., points at different radialdistances from the center of the substrate, the expected pressure can becalculated as a linear combination of the pressures in the chambers. Forexample, the estimated pressures can be calculated asEP _(x)=Σ_(i=1) ^(N) A _(i,x) *P _(i)  (12)where EP_(x) is the estimated pressure applied at a location x, P_(i) isthe pressure in the i^(th) chamber out of N chambers in the carrierhead, and A_(i,x) is a constant. As noted above, the location x can be aradial distance from the center of the substrate.

A relationship between the chamber pressures and the applied pressure,e.g., the values of the constants A_(i,x), can be determined based onmeasured data. In some examples, a pressure sensor (e.g., asubstantially planar sheet having an embedded one-dimensional ortwo-dimensional array of pressure sensors) can be inserted between thefront surface of the substrate 10 and a rigid support or a polishing pad110. At least one of the chambers 146 a-146 c is pressurized at abaseline pressure (e.g., between about 0-20 psi). Measurements providedby the measuring tool may reflect the downward pressure imparted on thesubstrate 10 at one or more positions (e.g., between about 1 and 100positions) across its front surface, when the baseline chamber pressureis applied. In some implementations, measurements are recorded atpositions regularly or irregularly spaced at radial distances from thecenter of the substrate 10. There will be more measurements thanindependently controllable chambers. In some implementations, themeasurements are recorded at positions below specific chambers. Forexample, a first set of multiple measurements may be recorded belowchamber 146 a and a second set of multiple measurements may be recordedbelow chamber 146 b.

In some implementations, chambers 146 a-146 c are pressurized one at atime to the baseline pressure while the other chambers are notpressurized. At this time measurements are recorded across the frontsurface of the substrate 10 (in some examples, the chambers arepressurized one at a time to different baseline pressures). The recordedpressure distributions based on each chamber may then be combined toprovide an aggregate pressure distribution reflecting the pressure spillover across zones of the front substrate surface (see FIG. 4). In someimplementations, the values of each constant A_(i,x), can be calculatedsimply by dividing the measured downward pressure at the particularpoint (from the pressure distribution profile) by the baseline pressurethat was applied to the chamber for the measurement.

In some examples, multiple pressure distributions for each chamber(recorded as described above) under various respective chamber pressurescan be measured. Collectively, the multiple pressure distributions canbe used in calculation of the estimated pressure profile. It may beadvantageous to record multiple pressure distributions at variousrespective chamber pressures for each chamber in order to increase theapproximation accuracy. FIG. 5 illustrates the pressure distributionsmeasured across the front face of a substrate at varying chamberpressures. As shown, pressure distributions were measured under abaseline chamber pressure, as well as chamber pressures 20% above andbelow than the baseline.

When multiple pressure distributions for each chamber are provided, theestimated pressures can be calculated asEP _(x)=Σ_(i=1) ^(N) A _(i,x) *P _(i) +B _(i,x)  (13)where EP_(x) where is the estimated pressure applied at a location x,P_(i) is the pressure in the i^(th) chamber out of N chambers in thecarrier head, and A_(i,x) is and B_(i,x) are constants. Again, thelocation x can be a radial distance from the center of the substrate.

In some implementations, the values of constants A_(i,x) and B_(i,x) canbe calculated by solving the following system of equations for eachchamber iMP _(i,x,1) =A _(i,x) *P _(i,1) +B _(i,x)  (14)MP _(i,x,2) =A _(i,x) *P _(i,2) +B _(i,x)  (15)where MP_(i,x,1) and MP_(x,2) are the measured pressures at a radiallocation x on the substrate when the chamber i is pressurized at a firstchamber pressure P₁ and second chamber pressure P₂ respectively.

In some implementations, one or more pressure distributions for eachchamber are measured while the other chambers are also pressurized. Forexample, FIG. 6 shows a collection of pressure distributions wherechamber 146 b is pressurized 20% above and below a baseline chamberpressure at which chambers 146 a and 146 c are pressurized. AlthoughFIG. 6 illustrates a uniform pressure, e.g., from each chamber at thesame pressure, the various chambers need not be at the same pressure(although the chambers should be pressurized), and the resultingpressure at the front surface of the need not be uniform. Without beinglimited by any particular theory, the degree to which the membranematerial defining a respective chambers will expand or bulge (and thusthe effect on pressure distribution) can depend on the pressure in theadjacent chamber. Thus, pressure distributions measured in this way canmore accurately reflect the variations in pressure at the front surfaceof the substrate due to changes in chamber pressure, since in operationthe chambers are typically at least partially pressurized.

For a given chamber, pressure distributions can be measured at one, two,or more than two chamber pressures (with more than two chamberpressures, the estimated pressure could be treated as a set ofindividual line segments, or a single linear function could be fit tothe data). The number of chamber pressures at which pressuredistributions are measured for a given chamber can be governed by abalance between available computing resources, accuracy requirements,and time for data collection.

In some implementations, one or more equations describing the expectedpressure profile in terms of the variable individual pressuredistributions for each chamber may be optimized by an iterative process(e.g., Newton's method of optimization) to closely approximate a targetpressure profile. The first step of the iterative process may be tocompare the pressure distribution profile measured at the baselinechamber pressure(s) to the target pressure profile. Then, based on thecomparison, at least one variable of the equation (e.g., a pressuredistribution based on one of the chambers) may be adjusted and anexpected pressure profile may be calculated based on the adjustedvariable. The expected pressure profile may then be compared to thetarget pressure profile, and another adjustment can be performed. Theiterative adjustments and comparisons may continue until an expectedpressure profile closely approximating the target pressure profile isdetermined (that is, until a threshold in the comparison value isreached).

In some examples the target pressure profile (TPP) can be described interms of a plurality of target pressures at respective radial locationson the substrate surface. Similarly, the estimated pressure profile(EPP) can be described in terms of a plurality of expected pressures atrespective radial locations on the substrate surface (i.e., EP_(x)—seeequations (12) and (13) above). As shown above, the expected pressure atany location can be calculated as a function of chamber pressures. Thus,a combination of chamber pressures for providing a suitableapproximation of the target pressure profile can be determined bysolving for respective chamber pressures that minimize a differencebetween the estimated pressure profile and the target pressure profile.That is, the value of EPP-TPP is minimized.

For example, a difference value between the estimated pressure profileand the target pressure profile, e.g., using sum of squares or absolutevalues, and the values of the chamber pressures can be calculated tominimize this difference. For example, this can be performed by usingone of the following equationsΔ=Σ_(x=1) ^(M)([Σ_(i=1) ^(N) A _(i,x) *P _(i) +B _(i,x) ]−TP_(x)])²  (16)Δ=Σ_(x=1) ^(M)([Σ_(i=1) ^(N) A _(i,x) *P _(i) ]−TP _(x)])²  (17)where Δ is the difference value, TP_(x) is the target pressure to beapplied at the x^(th) radial location out of M radial locations on thesubstrate, P_(i) is a pressure in the i^(th) chamber out of N chambersin the carrier head, and A_(i,x) and B_(i,x) are constants determinedbased on previously measured data. Values of P_(i) can be calculatedfind a minimum for Δ.

In some examples, an iterative process (e.g., Newton's method ofoptimization) may be used in conjunction with either of equations (16)or (17) to determine an appropriate combination of chamber pressures tofind a minimum difference between the estimated pressure profile and thetarget pressure profile, e.g., to find a minimum value for Δ. Forexample, a commercially available solver function may be used todetermine an appropriate combination of chamber pressures. In somecases, there may be several suitable combinations of chamber pressures.

Once an acceptable expected pressure profile is determined,interpolation and/or extrapolation techniques (other known mathematicalapproximation techniques may also be used) can be applied to themeasured baseline pressure distributions to determine a correspondingchamber pressure for each chamber. When using pressure distributionsfrom a single baseline pressure it may be necessary to assume a zeroscale pressure distribution (i.e., when the chambers are notpressurized, there is little or no polishing pressure applied).

Referring to FIGS. 7 and 8, example flow charts are illustrated. In somecases, the process steps shown in FIG. 7 may be performed by one or moreemployees of a manufacturer of the polishing apparatus 100. For example,one or more pressure distributions for each chamber may be measured andrecorded by the manufacturer (step 702), as described above. Thepressure distributions can be used to calculate (or otherwise determine)the relationships between chamber pressure and the downward pressure ofthe substrate on the polishing pad (step 704), as described above, e.g.,by calculating constants A_(i,x) and B_(i,x). This information may beprovided to a semiconductor fab by the manufacturer of the polishingapparatus 100.

In some cases, the process steps shown in FIG. 8 may be performed by oneor more employees of a semiconductor fab. For example, an appropriatetarget surface profile for a substrate may be determined (step 802) bythe semiconductor fab as part of a planarization technique for formingan integrated circuit. The initial surface profile of the substrate maybe measured (step 804). For example, the surface profile of thesubstrate can be measured at a metrology station prior to polishing, orcan be measured during polishing by an in-situ monitoring system. Therelationship between polishing rate and downward pressure of thesubstrate on the polishing pad can be determined (step 806). In somecases, this information can be obtained from data provided by a supplierof the polishing apparatus or polishing pad. In some other cases, thisrelationship can be empirically determined by the semiconductor fab,e.g., by experimentally determining the constant m (see equation (10)).In some implementations, the target pressure profile can be determined(step 808) based on the initial surface profile, the target surfaceprofile, and the relationship between polishing rate and pressure, asdescribed above. A suitable combination of chamber pressures to achievethe target pressure profile, and thus the target surface profile, maythen be determined (step 810) based on the known relationships betweenchamber pressure and the downward pressure on the substrate against thepolishing pad across the substrate surface, as described above.

As used in the instant specification, the term substrate can include,for example, a product substrate (e.g., which includes multiple memoryor processor dies), a test substrate, a bare substrate, and a gatingsubstrate. The substrate can be at various stages of integrated circuitfabrication, e.g., the substrate can be a bare wafer, or it can includeone or more deposited and/or patterned layers. The term substrate caninclude circular disks and rectangular sheets.

Embodiments of the invention and all of the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, or in computer software, firmware, or hardware, including thestructural means disclosed in this specification and structuralequivalents thereof, or in combinations of them. Embodiments of theinvention can be implemented as one or more computer program products,i.e., one or more computer programs tangibly embodied in an informationcarrier, e.g., in a machine-readable non-transitory storage device or ina propagated signal, for execution by, or to control the operation of,data processing apparatus, e.g., a programmable processor, a computer,or multiple processors or computers. A computer program (also known as aprogram, software, software application, or code) can be written in anyform of programming language, including compiled or interpretedlanguages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a computing environment. A computer program does notnecessarily correspond to a file. A program can be stored in a portionof a file that holds other programs or data, in a single file dedicatedto the program in question, or in multiple coordinated files (e.g.,files that store one or more modules, sub-programs, or portions ofcode). A computer program can be deployed to be executed on one computeror on multiple computers at one site or distributed across multiplesites and interconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application-specific integrated circuit).

The above described polishing apparatus and methods can be applied in avariety of polishing systems. Either the polishing pad, or the carrierhead, or both can move to provide relative motion between the polishingsurface and the substrate. For example, the platen may orbit rather thanrotate. The polishing pad can be a circular (or some other shape) padsecured to the platen. Some aspects of the endpoint detection system maybe applicable to linear polishing systems, e.g., where the polishing padis a continuous or a reel-to-reel belt that moves linearly. Thepolishing layer can be a standard (for example, polyurethane with orwithout fillers) polishing material, a soft material, or afixed-abrasive material. Terms of relative positioning are used; itshould be understood that the polishing surface and substrate can beheld in a vertical orientation or some other orientation duringoperation.

Particular embodiments of the invention have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results.

What is claimed is:
 1. A method of selecting values for a plurality ofindependently controllable chamber pressures of a carrier head in achemical mechanical polishing system, the carrier head including aplurality of chambers to apply the independently controllable chamberpressures to a plurality of corresponding zones on a substrate,comprising: for each chamber and corresponding zone, storing datarelating variation in removal profile on a front surface of thesubstrate to variation in the controllable chamber pressure of thechamber, the data including removal at a plurality of positions on thefront surface of the substrate, there being a greater number ofpositions than chambers, the plurality of positions including positionsinside the corresponding zone and positions outside the correspondingzone; determining a value of a chamber pressure for each chamber of theplurality of controllable chambers to minimize a difference between atarget removal profile and an expected removal profile calculated fromthe data relating variation in removal profile on a front surface of thesubstrate to variation in the chamber pressure for each chamber suchthat an expected removal rate for at least some positions within a zonefrom the plurality of zones includes an effect of the chamber pressurefor the chamber corresponding to the zone and an effect of the chamberpressure for a chamber that does not correspond to the zone; and storingthe value for each chamber pressure of the plurality of controllablechamber pressures.
 2. The method of claim 1, wherein the plurality ofzones are arranged concentrically and the plurality of positions areradial distances from the center of the substrate.
 3. The method ofclaim 2, wherein the plurality of positions include a first plurality ofpositions below a first zone of the plurality of zones and a secondplurality of positions below a second zone of the plurality of zones. 4.The method of claim 1, wherein the positions are regularly spaced acrossthe substrate.
 5. The method of claim 1, wherein storing data comprisesstoring a plurality of measured removal profiles, each measured removalprofile including an amount removed from a front surface of a substrateat each of a plurality of positions on the substrate.
 6. The method ofclaim 5, wherein the plurality of removal profiles includes a baselineremoval profile for one substrate polished under a set of baselinevalues for the plurality of controllable chamber pressures.
 7. Themethod of claim 6, wherein the plurality of removal profiles includes aplurality of adjusted removal profiles for additional substrates, eachadditional substrate is polished with exactly one of the plurality ofcontrollable chambers set to an adjusted value for the chamber pressurethat is different from the baseline value for the one substrate.
 8. Themethod of claim 7, wherein the one of the plurality of controllablechambers is different for each additional substrate.
 9. The method ofclaim 1, wherein determining a value for each chamber pressure comprisesiteratively adjusting at least one value, calculating the expectedremoval profile from the data, and determining a difference between anexpected pressure profile and a target pressure profile.
 10. The methodof claim 9, wherein determining the value for the chamber pressure foreach chamber using Newton's method of optimization.
 11. The method ofclaim 1, further comprising generating the target removal profile fromdata collected in-situ during polishing of a first substrate, andadjusting a pressure in a chamber during the polishing of the firstsubstrate to match the value for the chamber pressure for the chamber.12. The method of claim 1, further comprising generating the targetremoval profile from data collected in-situ during polishing of a firstsubstrate at a first platen, and polishing the first substrate at adifferent second platen using the chamber pressure for the chamber setto the value.
 13. The method of claim 1, further comprising generatingthe target removal profile from data collected in-situ during polishingof a first substrate at a first platen, and polishing a different secondsubstrate at the first platen using the chamber pressure for the chamberset to the value.
 14. A method of selecting values for a plurality ofcontrollable parameters of a chemical mechanical polishing system thatincludes a carrier head with a plurality of zones to apply independentlycontrollable pressures on a substrate, comprising: storing data relatingvariation in removal profile on a front surface of the substrate tovariation in the controllable parameters, the data including removal ata plurality of positions on the front surface of the substrate, therebeing a greater number of positions than chambers; wherein the storingdata further comprises storing a plurality of removal profiles, eachremoval profile including an amount removed from a front surface of asubstrate at each of a plurality of positions on the substrate, andwherein the plurality of removal profiles includes a baseline removalprofile for one substrate polished under a set of baseline values forthe plurality of controllable parameters; determining a value for eachparameter of the plurality of controllable parameters to minimize adifference between a target removal profile and an expected removalprofile calculated from the data relating variation in removal profileon a front surface of the substrate to variation in the controllable;and storing the value for each parameter of the plurality ofcontrollable parameters; wherein calculating the expected removalprofile comprises calculatingERP=BRP+[K][P′]*BRP where ERP is the expected removal profile, BRP isthe baseline removal profile, [K] is a matrix of constants having anumber of columns equal to the total number of parameters, N, and anumber of rows equal to the total number of radial positions on thesubstrate surface where the profile is measured, M, and [P′] is a columnmatrix of adjusted polishing system parameters.
 15. The method of claim14, wherein the plurality of controllable parameters include pressuresfor a plurality of chambers in the carrier head that apply pressure tothe plurality of zones.
 16. The method of claim 15, wherein theplurality of controllable parameters include a pressure for a chamber inthe carrier head that applies pressure to a retaining ring of thecarrier head.
 17. The method of claim 14, wherein the wherein theplurality of controllable parameters include a platen rotation rate or acarrier head rotation rate.
 18. The method of claim 14, wherein thereare a greater number of positions than parameters.
 19. The method ofclaim 14 wherein each parameter P_(i)′ of the column of adjustedpolishing system parameters [P′] comprises$P_{i}^{\prime} = \frac{P_{i} - {P\; 0_{i}}}{P\; 0_{i}}$ where P_(i) isthe value of the polishing parameter to be calculated, and P0_(i) is thebaseline value for the polishing parameter.
 20. The method of claim 19,wherein the matrix [K] can be expressed as $\begin{matrix}{\lbrack K\rbrack = \begin{bmatrix}K_{1,1} & \ldots & K_{1,N} \\\vdots & \ddots & \vdots \\K_{M,1} & \ldots & K_{M,N}\end{bmatrix}} & (6)\end{matrix}$ where M is the total number of radial substrate positionsx at which profile measurements are taken and N is the total number ofparameters.
 21. The method of claim 20, wherein constants of the matrixcomprise $\begin{matrix}{K_{x,i} = \frac{{ARP}_{x,i} - {BRP}_{x}}{{BRP}_{x}*\left\lbrack \frac{{PA}_{i} - {P\; 0_{i}}}{P\; 0_{i}} \right\rbrack}} & (5)\end{matrix}$ where K_(x,i) is the value of the matrix [K] correspondingto the substrate profile position x for the i^(th) parameter, ARP_(x,i)is the amount of material removed at radial position x for the i^(th)adjusted parameter, BRP_(x) is the amount of material removed at radialposition x according to the previously calculated baseline removalprofile, PA_(i) is the i^(th) parameter value used to generate thei^(th) adjusted removal profile, and P0_(i) is the i^(th) parametervalue for the baseline removal profile.